ONLINE APPENDICES. Paternalism, Libertarianism, and the Nature of Disagreement Uliana Loginova and Petra Persson

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1 ONLINE APPENDICES Paternalism, Libertarianism, and the Nature of Disagreement Uliana Loginova and Petra Persson B Appendix: Omitted Proofs Proof of Proposition 2. Preference disagreement. by a corresponding lemma. We consider each type of disagreement in turn. The proof consists of two steps, each described Lemma 1. For any cost of coercion q 0 and preference distribution f(b) there exists a threshold ϕ TT (q, b, b 2 ), s.t. a FRE exists if and only if ϕ ϕ TT (q, b, b 2 ). Proof. Assume that the authority gets the signal s. If the authority coerces, she chooses the action a that maximizes her expected utility given by: + E(U A (a, θ) s) = p(s) [(a 1) 2 + ϕ(a 1 b i ) 2 ]f(b i )db i (1 p(s)) + Thus, the authority optimally imposes the action: a A (s) =p(s)+ [a 2 + ϕ(a b i ) 2 ]f(b i )db i q. ϕ 1+ϕ b. Her expected utility from coercion is E(U A (a, θ) s) = p(s) (1 + ϕ)(a A (s) 1) 2 2ϕ b(a A (s) 1) + ϕb 2 (1 p(s)) (1 + ϕ)(a A (s)) 2 2ϕ ba A (s)+ϕb 2 q = (1 + ϕ)e (a A (s) θ) 2 s +2ϕ b [a A (s) p(s)] ϕb 2 q 2 ϕ = (1 + ϕ) + p(s)(1 p(s)) +2 1+ϕ b 1+ϕ b ϕ2 2 ϕb 2 q = ϕ 2 1+ϕ b 2 (1 + ϕ)p(s)(1 p(s)) ϕb 2 q. 41

2 If instead the authority communicates some message m to the population, then the population s posterior becomes ˆp = p(m) (bysymmetry,inanyequilibrium all individuals hold the same posteriors after observing the authority s message). Hence, each individual i picks his preferred action a i (m) =ˆp + b i, which generates the following expected payoff to the authority: + E (U A (a(m), θ) s) = p(s) (ˆp + b i 1) 2 f(b i )db i + ϕ(ˆp 1) 2 + (1 p(s)) (ˆp + b i ) 2 f(b i )db i + ϕˆp 2 = p(s) (1 + ϕ)(ˆp 1) 2 +2 b(ˆp 1) + b 2 (1 p(s)) (1 + ϕ)ˆp 2 +2 bˆp + b 2 = (1 + ϕ)(ˆp 2 2ˆpp(s)+p(s)) 2 b(ˆp p(s)) b 2 = (1 + ϕ)(ˆp p(s)) 2 (1 + ϕ)p(s)(1 p(s)) 2 b(ˆp p(s)) b 2. Communication generates a greater expected payoff to the authority than coercion if ϕ 2 1+ϕ b 2 ϕb 2 q (1 + ϕ)(ˆp p(s)) 2 2 b(ˆp p(s)) b 2. (B.1) In the particular case of truthful advising this reduces to ϕ 2 1+ϕ b 2 ϕb 2 + b 2 q 0, ϕ 2 (b 2 b 2 ) ϕq + b 2 q 0. (B.2) The parabola in the LHS is concave and achieves its maximum at ϕ 0. This implies that the LHS decreases in ϕ for ϕ 0. In particular, the authority prefers to act as a truth telling advisor if and only if the level of altruism is sufficiently large: ϕ ϕ TT (q, b, b 2 ). The threshold ϕ TT (q, b, b 2 ) 1ifthe average bias is sufficiently small: b 2 2 q 0. If coercion is costless, q =0,thentruthfuladvisingispreferredforsufficiently large ϕ, for any non-degenerate distribution ( b 2 = b 2 ), i.e., ϕ TT (0, b, b 2 ) < + ; while coercion is always preferred for any degenerate distribution ( b 2 = b 2 ), i.e., ϕ TT (0, b, b 2 )=+. Clearly, ϕ TT (q, b, b 2 ) strictly exceeds 0 and decreases with q for q<b 2,and ϕ TT (q, b, b 2 )=0whenq b 2 42

3 Lemma 2. For every f(b) and q<q C ( b, b 2 ) there exists a threshold ϕ C (q, b, b 2 ) > 0, s.t. for any ϕ < ϕ C (q, b, b 2 ) every equilibrium outcome involves the authority imposing her preferred action with non-zero probability. Moreover, for every b and q<q CC ( b, b 2 ) there exists a threshold ϕ CC (q, b, b 2 ) > 0, s.t. for any ϕ < ϕ CC (q, b, b 2 ) there is a unique equilibrium, in which the authority coerces the individual for each signal s {0, 1}. Proof. As regards the first part of the Lemma, we show that sufficiently low cost q and altruism ϕ preclude the existence of communication equilibria. We study two different types of communication equilibria: (i) only one posterior belief ˆp is induced, and (ii) two different posterior beliefs ˆp 0 < ˆp 1 are induced. In case (i), the authority s communication yields the same posterior belief in the population for both signals. Thus, no informative communication takes place and the posterior belief necessarily coincides with the prior, ˆp = π. This means that each individual i picks an action a i = π+b i. Such a communication equilibrium fails to exist when, after some signal s, the authority would rather exercise her power to choose her preferred action, i.e., when the inequality (B.1) holdswiththeoppositesign: ϕ 2 1+ϕ b 2 ϕb 2 q> (1 + ϕ)(π p(s)) 2 2 b(π p(s)) b 2. Assume that the authority does not incorporate the individual s payoff into her utility: ϕ =0. Then,forsufficiently low cost q<max s [(π p(s)) 2 +2 b(π p(s)) + b 2 ], the authority coerces the individual for at least one signal s. Now consider case (ii), in which the induced posterior beliefs of the population are different for different signals, ˆp 0 < ˆp 1. The authority cannot be indifferent between these beliefs after both signal realizations. Thus, for these actions to be induced in equilibrium, it must be the case that the authority mixes between messages for no more than one signal. This implies that ˆp s = p(s) foratleastonesignals. Hence, a non-altruistic authority who obtained the signal s will choose to impose her preferred action whenever the inequality (B.2) breaksdown,i.e.,whenq<b 2. Clearly, she will still do so for sufficiently low levels of altruism. Together, cases (i) and (ii) imply that for a sufficiently low coercion cost, q<q C ( b, b 2 ), and altruism level, ϕ < ϕ C (q, b, b 2 ), each equilibrium involves coercion with a non-zero probability. Clearly, ϕ C (q, b, b 2 )decreaseswithq for q<q C ( b, b 2 ). As regards the second part of the Proposition, we argue that for sufficiently lowq and ϕ no communication is sustainable in equilibrium. Below we consider the two possibilities of how communication can arise. 43

4 First, assume that the authority communicates with positive probability for both signal realizations s =0, 1. If only one population s posterior ˆp [p(0),p(1)] is induced in equilibrium, then a non-altruistic authority would strictly prefer to coerce for at least one signal when q<q(ˆp, b, b 2 ) = max[(ˆp p(s)) 2 +2 b(ˆp p(s)) + b 2 ]. s (B.3) Clearly, q(ˆp, b, b 2 ) > 0foreveryˆp. Because q(ˆp, b, b 2 ) is a continuous function of ˆp, there exists 0 < q( b, b 2 ) = minˆp [p(0),p(1)] q(ˆb, b, b 2 ). The inequality (B.3) is strict for any q< q( b, b 2 )andˆp [p(0),p(1)]; hence, the authority still prefers to coerce whenever the level of altruism is below the threshold ϕ(q, ˆp, b, b 2 ) > Because ϕ(q, ˆp, b, b 2 )iscontinuousinˆp [p(0),p(1)], there exists 0 < ϕ(q, b, b 2 )= min ˆp [p(0),p(1)] ϕ(q, ˆp, b, b 2 ). As a result, for q< q( b, b 2 )andϕ < ϕ(q, b, b 2 ), there exists no equilibrium with non-zero communication after both signals and only one induced posterior belief. Next, if two different posterior beliefs ˆp 0 < ˆp 1 are induced in equilibrium, then the authority strictly prefers inducing one action over the other for at least one signal, meaning that ˆp s = p(s) for some s {0, 1}. Clearly, such an equilibrium cannot exist for q lower than b 2 and sufficiently low ϕ. Second, suppose that the authority communicates with a non-zero probability for one signal and coerces with certainty for the other signal. For example, assume that the authority sometimes communicates after s =1and always coerces after s =0. Inthiscase,themerefactthattheauthoritydidn t impose an action reveals to the population that the signal is s =1. As a result, the induced population s posterior ˆp necessarily equals p(1). Such an equilibrium fails to exist for q lower than b 2 and sufficiently low ϕ. The case when the authority sometimes communicates the low signal s = 0andalwayscoerces when s =1isanalogous. Itfollowsthatforeveryq<b 2, communicating after only one signal cannot be an equilibrium if ϕ is sufficiently low. Combining the results of the two cases, when the coercion cost q and altruism level ϕ are below the respective bounds q CC ( b, b 2 )andϕ CC (q, b, b 2 ), the unique equilibrium is the one where the authority always coerces. 32 Assuming that the maximum in (B.3) is achieved at ŝ, the threshold ϕ(q, ˆp, b, b 2 )is ϕ determined as a closest to 0 positive root of 1+ϕ b 2 2 ϕb 2 q = (1 + ϕ)(ˆp p(s)) 2 2 b(ˆp p(s)) b 2. 44

5 Denoting q = q CC ( b, b 2 ), Lemma 1 and Lemma 2 imply the preference disagreement result of Proposition 2. Evidently, ϕ CC (q, b, b 2 ), ϕ C (q, b, b 2 ), and ϕ TT (q, b, b 2 )aredecreasinginq. Clearly, a FRE is preferred by the individuals to any other (pure or mixed strategies) PBE. Now we show that a FRE is preferred by the authority to any other pure strategies PBE when ϕ exceeds ϕ TT (q, b, b 2 ). First, the FRE is always preferred to purely uninformative advising, where the same message is sent after both signals. Indeed, the FRE generates a greater expected utility to the advisor whenever (πγ +(1 π)(1 γ)) (1 + ϕ)(π p(1)) 2 2 b(π p(1)) + (π(1 γ)+(1 π)γ) (1 + ϕ)(π p(0)) 2 2 b(π p(0)) 0. This inequality can be rewritten as (1 + ϕ) (πγ +(1 π)(1 γ))(π p(1)) 2 +(π(1 γ)+(1 π)γ)(π p(0)) 2 0, which satisfied for all ϕ. Second, the FRE is preferred to equilibria where the authority coerces after some or both signals. Indeed, as the previous analysis illustrates, after any signal s the authority prefers truth-telling over coercion for ϕ ϕ TT (q, b, b 2 ). Opinion disagreement. The proof proceeds in two steps; each step is formulated as a corresponding lemma. Lemma 3. For any opinion distribution g / G, there exists a threshold ϕ TT (q, g), s.t. a FRE exists if and only if ϕ ϕ TT (q, g). Proof. Consider some opinion distribution g(π) / G. That is, truthful reporting to the population described by g(π) isincentivecompatiblewhenc =0, given that the individuals believe the reported message. Truth-telling is preferred to coercion after signal s if ϕ 1 0 pa (p i 1) 2 +(1 p A )(p i ) 2 1 q ϕ pa (p A 1) 2 +(1 p A )(p A ) 2 g(π i )dπ i, where p A = p A (s) andp i = p i (s) =a i (s) aretheauthority sandindividuali s posterior beliefs that θ = 1afterthesignals. This condition can be rewritten as ϕ pa (p i 1) 2 +(1 p A )(p i ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i q. π i =

6 The integral in the LHS is strictly positive because p A (p i 1) 2 +(1 p A )(p i ) 2 > p A (p A 1) 2 +(1 p A )(p A ) 2 for any p i = p A (i.e., π i =0.5) and π i =0.5 g(π i)dπ i > 0. Thus, a FRE exists if and only if ϕ is below a threshold ϕ TT (q, g). Note that ϕ TT (q, g) > 0whenq>0andϕ TT (0,g)=0. Lemma 4. For any opinion distribution g(π) and q 0, there exist thresholds ϕ C (q, g) ϕ CC (q, g). For ϕ > ϕ CC (q, g), there exists a unique equilibrium in which the authority coerces with probability one after each signal s. For ϕ > ϕ C (q, g), every equilibrium involves the authority coercing with strictly positive probability. Proof. First, we prove that all equilibria with communication after both signal realizations break down for a sufficiently high level of altruism ϕ > ϕ C (q, g). To show this, we distinguish between (i) communication equilibria where each individual i takes only one action â i, and (ii) communication equilibria where each individual i takes two different actions â i,0 < â i,1 with positive probabilities. In case (i), communication is necessarily uninformative, and hence, each individual i chooses an action equal to his prior belief, â i = π i. Coercion after some signal s is preferred to such communication whenever ϕ pa (π i 1) 2 +(1 p A )(π i ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i q, π i =p A where p A = p A (s). The integral in the LHS is strictly positive, because p A (π i 1) 2 +(1 p A )(π i ) 2 >p A (p A 1) 2 +(1 p A )(p A ) 2 for any π i = p A and π i =p A g(π i )dπ i > 0. Thus, coercion is strictly preferred for sufficiently large ϕ, so such a communication equilibrium breaks down. In case (ii), the communicating authority induces the individuals to take different actions â i,0 < â i,1. Clearly, rational behavior on the part of each individual i ensures that â i,0, â i,1 [p i (0),p i (1)]. Because g(π)dπ > 0and π= =π A is the median of g(π), w.l.o.g. we assume that g(π)dπ > 0 π>0.5+ε for some ε > 0. Coercion after signal s =0ispreferredtosuchcommunication whenever max ϕ m {0,1} 1 0 pa (â i,m 1) 2 +(1 p A )(â i,m ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i q, where p A = p A (0). Now we show that the integral in the LHS has a lower 46

7 bound δ > 0thatisindependentoftheparticularactionsâ i,0 and â i,1. Indeed, max m {0,1} 1 π i >0.5+ε π i >0.5+ε = δ > 0, 0 pa (â i,m 1) 2 +(1 p A )(â i,m ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i pa (â i,0 1) 2 +(1 p A )(â i,0 ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i pa (p 0.5+ε (0) 1) 2 +(1 p A )(p 0.5+ε (0)) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i where p 0.5+ε (0) is the posterior of the individual with the prior 0.5 +ε after getting the signal s =0. 33 Here the first inequality holds because the expression inside the integral always is positive and for every i the action a i,0 is closer to the advisor s preferred action p A than a i,1 is. The second inequality holds because p A (0) <p 0.5+ε (0) <a i,0 for any communication equilibrium of this type. As a result, the authority prefers coercion over communication when her altruism level exceeds some threshold (which is independent of particular actions â i,0 and â i,1 ). Combining the results of the two cases, we obtain that for a sufficiently high level of altruism, ϕ > ϕ C (q, g) (where the subscript C denotes coercion), the authority cannot communicate with probability one in equilibrium. Second, we argue that for a sufficiently large ϕ no communication is sustainable in equilibrium. In order to do this, we consider two manners in which information transmission can arise: (i) the authority communicates with positive probability after both signal realizations, and (ii) the authority communicates with non-zero probability after one signal realization and coerces with certainty after the other. In case (i), we consider two possibilities: either each individual i takes only one action â i, or each individual i takes two different actions â i,0 < â i,1 with positive probabilities. In the first case the authority prefers to coerce after the signal s =0if ϕ 1 0 pa (â i 1) 2 +(1 p A )(â i ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i q, where p A = p A (0), and we continue to assume that g(π)dπ > 0for π>0.5+ε some ε > 0. As it was shown, the integral in the LHS exceeds some δ > 0that 33 Note that the individual with the prior 0.5+ε is hypothetical, i.e., he might not be a part of the population. 47

8 is independent of particular actions â i. Hence, for ϕ exceeding some threshold (which is independent of particular actions â i ), there exists no equilibrium with non-zero communication for both signals where each individual i takes only one action. Second, if every individual i takes different actions â i,0 < â i,1, then, by an analogous analysis, the authority will prefer to coerce for sure when ϕ is greater than some threshold (which is independent of particular actions â i,0 and â i,1 ). In case (ii), suppose, for example, that the authority coerces for sure after getting the signal s = 0. In this equilibrium, the authority s decision to communicate perfectly reveals that she observed the signal s = 1, so the individuals implement a i (1) = p i (1). Clearly, for a high enough level of altruism ϕ pa (p i 1) 2 +(1 p A )(p i ) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i >q. π i =0.5 where p A = p A (1) and p i = p i (1), and the authority would choose to improve upon this action; consequently, such an equilibrium breaks down. Combining the results of (i) and (ii), for sufficiently high ϕ > ϕ CC (q, g) the unique equilibrium is the one where the authority always coerces the individual. Note that ϕ CC (q, g) ϕ C (q, g) ϕ TT (q, g) > 0whenq>0and ϕ CC (0,g)=ϕ C (q, g) =0. Lemma 3 and Lemma 4 imply the opinion disagreement result of Proposition 2. Evidently, ϕ CC (q, g), ϕ C (q, g), and ϕ TT (q, g) areincreasinginq. QED. Proof of Proposition 3. The case of preference disagreement is considered in the main text; here we present a proof for the opinion disagreement case. Consider some distribution g(π) G and some cost of lying c>0. By Proposition 1, the advisor can be credible (to the entire population) under public communication iff ϕ ϕ(c, g). At the same time, under targeted communication, the advisor can report truthfully to the part of population with sufficiently close opinions. Thus, the advisor is (weakly) more credible with public messages when ϕ ϕ(c, g), and is (weakly) more credible with private messages when ϕ > ϕ(c, g). Assume that ϕ(c, g) > 0andthateveryindividualbelievesthemessage he gets (either public or private). If ϕ = ϕ(c, g), then the advisor can be credible (to the entire population) in public communication case, but can not be credible to all individuals with private messages. Indeed, under public communication, the advisor is indifferent between reporting some signal s 48

9 {0, 1} truthfully and lying (the advisor weakly prefers to report the other signal truthfully). That is, the benefit from lying about s to extreme individuals of measure η equals the loss from lying to the other 1 η less extreme individuals plus the cost c. This implies that there is a set of individuals of measure η, 0 < η η such that the benefit of lying to these individuals exceeds c. Hence, under targeted communication, the advisor will strictly prefer to misreport the signal s to the η extreme individuals. Now consider some distribution g(π) / G. Then for any cost of lying c 0andaltruismlevelϕ 0 public communication is credible. At the same time, the advisor can be non-credible to some individuals with private messages. Thus, for all ϕ 0theadvisoris(weakly)morecredibleunder public communication. Clearly, all individuals prefer a FRE to any other (pure or mixed strategies) PBE. Now we show that in private communication with individual i, a FRE is also preferred by the advisor to any other pure strategies PBE for any π i (0, 1). Hence, a FRE is preferred under public communication as well; more credible communication is always beneficial to an advisor with prior π A =0.5 (relativetolesscrediblecommunication). Assume that π i > 0.5 (thecaseofπ i < 0.5 issimilarandthecaseof π i =0.5 is straightforward). In a FRE, individual i takes actions p 0 = p i (0) and p 1 = p i (1) after messages 0 and 1, respectively. In the uninformative equilibrium the advisor sends the same message independently of the signal, and i takes the action π = π i. The FRE generates a greater expected utility to the advisor whenever 1 (1 γ)(p0 1) 2 γp 2 0 +(1 γ)(π 1) 2 + γπ 2 2 > 1 γ(p1 1) 2 +(1 γ)p 2 1 γ(π 1) 2 (1 γ)π 2. 2 This inequality can be rewritten as (π p 0 )(π + p 0 2(1 γ)) > (p 1 π)(π + p 1 2γ). First, note that π p 0 >p 1 π > 0, i.e., the individual with prior π > 0.5 is more responsive to a low signal s =0thantoahighsignals = 1. Second, it is easily verified that p 1 p 0 < 2γ 1(theindividualwithpriorπ > 0.5 is on average less responsive than is the individual with prior 0.5). Hence, π + p 0 2(1 γ) > π + p 1 2γ, which by π + p 0 2(1 γ) > 0 implies that the FRE in private communication with i is preferred to the uninformative equilibrium. QED. 49

10 C Appendix: Dominance solvable setting C.1 The altruistic advisor Here we consider the following modification of the baseline model presented in Section 3. In the first stage of the game, a signal about the state s, with precision Pr(s = θ θ) γ (0.5, 1), is realized. This signal is privately observed by the advisor with probability α (0, 1); the advisor remains uninformed with the probability (1 α). Whether the advisor observes the signal is not observable by the population. If the advisor obtains the signal, she can either pass the signal s on to all individuals, or hide it at a cost c 0, in which case the individuals learn nothing about the state. We denote this case by. That is, the advisor can incur a costly effort to hide information from the population. Importantly, when the individual observes, he cannot tell whether the advisor (i) did not receive a signal or whether she (ii) obtained the signal but chose to hide it. Upon learning the signal s or observing no information, each individual chooses an action a R. The assumptions about the individuals preferences, opinions, and payoffs are the same as in the baseline model, and the advisor s utility is given by U A (a, θ) = 1 0 (a i θ) 2 di ci A hides info ϕ 1 0 (a i θ b i ) 2 di. Consider the action that individual i optimally chooses given the available information. In the case when individual i gets to know the signal, his action is a i (1) = p i (1) + b i if s = 1 and a i (0) = p i (0) + b i if s = 0. When no information is revealed to i, he rationally chooses a i ( ) =p i ( ) +b i, where p i ( ) [p i,0, p i,1 ] (p i (0),p i (1)), where the upper bound p i,1 corresponds to i s posterior given that the advisor passes on the signal s =0andincurseffort to hide s =1. Similarly,thelowerboundp i,0 is i s posterior when the advisor reveals s =1andincurseffort to hide s =0. Theexpressionsforp i,1 and p i,0 are given by: p i,1 = p i,0 = π i (1 α + αγ) π i (1 α + αγ)+(1 π i )(1 α + α(1 γ)), π i (1 α + α(1 γ)) π i (1 α + α(1 γ)) + (1 π i )(1 α + αγ). Preference disagreement. Under preference disagreement, if the level of altruism is sufficiently large, the advisor passes on the signal to the population whenever she has one. 50

11 Lemma 5. For any cost c 0 and preference distribution f(b) with mean b there exists a ϕ(c, b), s.t. for every ϕ > ϕ(c, b) there is a unique rationalizable outcome in which the advisor reveals the signal if she has one. Proof. Assume that the population s belief after observing no signal is given by some p( ) [p 0, p 1 ]. If the informed advisor does not hide the signal s =1, she gets an expected payoff of: E [U A (a(1), θ) s =1] = p(1) (p(1) + b 1) 2 + ϕ(p(1) 1) 2 (1 p(1)) (p(1) + b) 2 + ϕp(1) 2 + b 2 b 2. If the advisor exerts an effort to hide s =1,herexpectedpayoff is given by E [U A (a( ), θ) s =1] = p(1) (p( )+ b 1) 2 + ϕ(p( ) 1) 2 (1 p(1)) (p( )+ b) 2 + ϕp( ) 2 + b 2 b 2 c. The advisor prefers to pass the signal s =1ontothepopulationwhenever p(1) (p(1) + b 1) 2 + ϕ(p(1) 1) 2 (1 p(1)) (p(1) + b) 2 + ϕp(1) 2 > p(1) (p( )+ b 1) 2 + ϕ(p( ) 1) 2 (1 p(1)) (p( )+ b) 2 + ϕp( ) 2 c. This condition is equivalent to the one when the advisor communicates to asingleindividualwithpreferencebias b. Because p( ) <p(1), it can be rewritten as 2 b ϕ > 1+ p(1) p( ) c (p(1) p( )). 2 Similarly, using p( ) >p(0), the advisor s incentive compatibility condition to pass on the signal s =0tothepopulationcanbewritten ϕ > 1 2 b p( ) p(0) c (p(0) p( )). 2 Because p( ) [p 0, p 1 ] (p(0),p(1)), for sufficiently large ϕ > ϕ(c, b) itisa dominant strategy for the informed advisor to reveal the signal independently of p( ). Opinion disagreement. Under opinion disagreement, there exists a nondegenerate set of distributions for which the the advisor hides some signal when the level of altruism is sufficiently large. 51

12 Lemma 6. There exists a non-degenerate set of opinion distributions G such that hiding some signal is a strictly dominant strategy for the advisor when c =0. For any g(π) G and c>0, there exists a threshold level of altruism ϕ(c, g) such that the unique rationalizable outcome involves the advisor hiding some signal for sure when ϕ ϕ(c, g). Proof. First, we show that G is non-empty and uncountable. Consider a simple opinion distribution with two types of individuals, t 1 and t 2, with priors π 1 < π A =0.5 < π 2. The types are equally prevalent, i.e., g(π 1 )=g(π 2 )=0.5. Assume that type t 1 is extreme and is not responsive to the signal s, π 1 =0. Assume that π 2 is sufficiently high to ensure that any p( ) [p 2,0, p 2,1 ] is closer to p A (1) = γ than p 2 (1). This implies that in the case of c =0itisastrictly dominant strategy for the advisor to always hide the signal s =1. Clearly, G contains all such (and, by continuity, many other) distributions; hence, it is uncountable. Second, we note that for any g(π) G and c>0, the advisor s strictly dominant strategy is to hide some signal when ϕ is sufficiently large. Indeed, for any given cost of lying, increasing the level of altruism makes the net benefit from hiding the signal larger relative to the cost c. C.2 The altruistic authority After observing the signal, s, or after observing no signal,, the authority(a, she) chooses between sending a public message m, after which the individuals choose actions, and mandating an action for all individuals, a A R. As before, the cost of coercion is q, where q c =0. Preference disagreement. Under preference disagreement, for sufficiently high levels of altruism, the authority never coerces or hides information. Lemma 7. For any for any preference distribution f(b) with b 2 = b 2 and any q 0, there exists a threshold ϕ(q, b 2, b 2 ), such that for any ϕ > ϕ(q, b 2, b 2 ) there is a unique rationalizable outcome where the authority never coerces the individuals and reveals the signal if she has one. Proof. Assume that the authority obtained the signal. By Lemma 5, for sufficiently large ϕ > ϕ(c =0, b) theauthorityalwaysprefersrevealingthesignal to hiding it independently of p( ) [p 0, p 1 ]. Next, comparing signal revelation 52

13 to coercing and imposing a A (s) =a A (s) =p(s) + 1+ϕ b, ϕ signal revelation is preferred when: ϕ 2 (b 2 b 2 ) ϕq + b 2 q 0. (C.1) Thus, for sufficiently large ϕ > ϕ(q, b 2, b 2 ) ϕ > ϕ(c =0, b) itisadominant strategy of the informed authority to pass on the signal and never coerce. Now consider an uninformed authority and assume that ϕ > ϕ(q, b 2, b 2 ). If the authority does nothing, each individual i rationally chooses action π + b i (the individuals know that the authority never hides information, because ϕ > ϕ 1 (q, b 2, b 2 )). If the authority coerces, she chooses a A ( ) =π + 1+ϕ b. ϕ Thus, the authority and the individuals hold the same beliefs about the state, in which case the authority prefers not to coerce whenever (C.1) issatisfied. As a result, given rational behavior on the part of the population, whenever ϕ > ϕ(q, b 2, b 2 )itisastrictlydominantstrategyfortheauthoritynottocoerce the individuals, and to instead pass the signal on whenever she is informed. Opinion disagreement. Under opinion disagreement, for sufficiently high levels of altruism, the authority coerces the population. Lemma 8. For any for any opinion distribution g(π) and any q 0, there exists a threshold ϕ(q, g), such that for any ϕ > ϕ(q, g) there is a unique rationalizable outcome where the authority coerces the individuals with probability one. Proof. W.l.o.g. we assume that g(π)dπ > 0. Consider the authority π>0.5+ε who obtained the signal s = 0. Iftheauthorityrevealsthesignaltothe population, every individual i optimally chooses an action p i (0) = p A (0) if π i =0.5. Thus, for sufficiently large ϕ > ϕ 0 (q, g) thebenefitfromcoercing exceeds the cost and the authority finds it strictly optimal to impose her preferred action p A (s). If the authority hides the signal s =0anddoesnot coerce, then each individual i picks an action p i ( ) [p i (0),p i (1)]. The benefit from coercing after signal s =0is 1 ϕ ϕ 0 pa (p i ( ) 1) 2 +(1 p A )(p i ( )) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i π i >0.5+ε δ 0 > 0, pa (p i ( ) 1) 2 +(1 p A )(p i ( )) 2 p A (p A 1) 2 (1 p A )(p A ) 2 g(π i )dπ i where p A = p A (0). Here the first inequality holds because the expression inside the integral always is positive, and the second inequality holds because 53

14 for every i the action p i ( ) p 0.5+ε ( ) >p A (0) when π i > 0.5+ε. Thus, for ϕ > ϕ 1 (q, g) ϕ 0 (q, g) theauthoritystrictlypreferstocoercethepopulation after observing the signal s =0independentlyofp i ( ). This, in turn, implies that p i ( ) [π i,p i (1)]. Assume that ϕ > ϕ 1 (q, g). If an uninformed authority coerces the population, she derives an expected benefit 1 ϕ ϕ 0 πa (p i ( ) 1) 2 +(1 π A )(p i ( )) 2 π A (π A 1) 2 (1 π A )(π A ) 2 g(π i )dπ i π i >0.5 δ 1 > 0. πa (p i ( ) 1) 2 +(1 π A )(p i ( )) 2 π A (π A 1) 2 (1 π A )(π A ) 2 g(π i )dπ i Here the first inequality holds because the expression inside the integral always is positive, and the second inequality holds because for every i the action p i ( ) π i > π A when π i > 0.5. Thus, the authority strictly prefers to coerce after observing if ϕ > ϕ 2 (q, g) ϕ 1 (q, g). Assume that ϕ > ϕ 2 (q, g) andthattheauthorityobservedthesignals =1. By the same logic as above, she would strictly prefer coercion to revealing the signal when ϕ > max{ϕ 2 (q, g), ϕ 3 (q, g)}. If she hides the signal and does not coerce, then each individual i optimally chooses p i ( ) =p i (1) (because if the individuals get to choose the action, it necessarily means that s =1, provided that ϕ > ϕ 2 (q, g)). This is strictly dominated by coercion for ϕ > max{ϕ 2 (q, g), ϕ 3 (q, g)}. As a result, when ϕ > ϕ(q, g), the unique rationalizable outcome is the one in which the authority always coerces the individuals. D Appendix: Setting with one individual D.1 Model We first develop a framework where an individual obtains information from an altruistic advisor before making a decision. We then consider an alternative framework where theadvisor in fact is anauthority, who can communicate with the individual before he makes his decision, but who also can choose to simply coerce the individual to take a certain action. In both of these frameworks, we analyze the impact of altruism under two different assumptions on the 54

15 nature of disagreement between the two parties: conflicting preferences and conflicting opinions. D.1.1 The altruistic advisor There are two players, an individual (I, he) and an advisor (A, she). The individual must take an action, a R. His payoff from the action depends on an unknown state of the world, θ {0, 1}. While he is unable to obtain information about the state by himself, he can rely on the privately informed advisor for such information. The players prior beliefs about the state of the world are characterized by Pr i (θ =1)=π i (0, 1), for i {I,A}. 34 Following Che and Kartik (2009), we say that the players have different opinions when π I = π A. In the first stage of the game, the advisor privately observes a signal about the state, s, with precision Pr(s = θ θ) γ (0.5, 1). Second, she sends a message m {0, 1} to the individual. If the advisor lies, i.e., sends message m = s, she incurs a cost c Third, upon receiving the advisor s signal, the individual chooses an action a R. The players material payoffs fromthisactionaregivenbyu I (a, θ) = (a θ b(θ)) 2 and u A (a, θ) = (a θ) 2 ci {m=s}, where the bias b(θ) 0captures their preference disagreement, and I {m=s} is an indicator variable taking the value one if and only if the advisor lies, and zero otherwise. In this standard model of communication, we allow the advisor to be altruistic, i.e., the players utilities are given by U I (a, θ) = u I (a, θ) = (a θ b(θ)) 2, U A (a, θ) = u A (a, θ)+ϕu I (a, θ) = (a θ) 2 ci {m=s} ϕ(a θ b(θ)) 2, where ϕ 0capturesthedegreetowhichtheadvisorcaresaboutthe(material) well-being of the individual. 36 The prior beliefs π I and π A, the signal precision γ, the lying cost c, the preference alignment b(θ), and the degree of altruism ϕ are common knowledge. 34 Given that the state is binary, all beliefs can be characterized by Pr(θ = 1). 35 This formulation encompasses cheap-talk (c = 0) and hard information (c = ). We maintain that c (0, ) unless we explicitly consider one of these two cases. 36 We show in Section D.5 that all the main results remain valid if friendship is mutual. We choose unidirectional friendship as our baseline formulation not only for simplicity, but also because it better reflects a social planner who cares about a citizen. 55

16 Strategies and equilibrium. A pure strategy of the advisor specifies, for each signal s, the message m(s) thatshesends,m : {0, 1} {0, 1}. The individual s posterior beliefs conditional on message m are described by Pr I (θ = 1 m), where superscript I signifies that the individual forms his beliefs using his prior π I. A pure strategy of the individual specifies, for each message m, action a I (m) thathetakes,a I : {0, 1} R. We use a solution concept of Perfect Bayesian Equilibria (PBE), where the advisor maximizes her expected utility for each signal s given the individual s strategy a I (m); the individual maximizes his expected utility given his beliefs Pr I (θ =1 m) aftereachmessage m; and the beliefs Pr I (θ =1 m) satisfybayes rulewheneverpossible. D.1.2 The altruistic authority We now replace the advisor with an authority. The authority can choose to behave like an advisor that is, to provide the individual with information before he chooses a R himself but the authority also has the option to instead impose an action on the individual. Formally, after observing the signal, s, the authority chooses between sending a message m(s) totheindividual(inwhich case the game proceeds as in the altruistic advisor framework), and engaging in coercion, whereby the authority simply picks her desired action a R for the individual. 37 We assume that the authority must incur a cost q>0ifshecoercesthe individual. Depending on the application, this cost may reflect the instrumental cost of active intervention, or the authority s intrinsic aversion against removing the individual s liberty to choose. In this altruistic authority framework, we let c =0. Thisreflectsthefact that in many of the applications we consider, it is reasonable to assume that the lying cost is negligible relative to the cost of coercion. 38 This is merely a simplification; all insights remain valid for c>0. The parties utilities are thus 37 Note that such coercion differs from delegation, whereby the individual decides that the authority should exercise the action choice. In contrast, under coercion, the authority herself decides when she should exercise the action choice on behalf of the individual. Moreover, because the authority observes the signal s before deciding whether to coerce, whereas the individual does not observe the signal s, the coercion choice is taken ex interim, whereas a delegation choice must be made ex ante. We discuss how these concepts are related in Section D This is discussed in depth in Section D.3. 56

17 given by U I (a, θ) = u I = (a θ b(θ)) 2, U A (a, θ) = u A + ϕu I = (a θ) 2 qi {coercion} ϕ(a θ b(θ)) 2, where I {coercion} takes value one if and only if coercion occurs. Strategies and equilibrium. A pure strategy of the authority specifies, for each signal s, whether she chooses to coerce or not coerce the individual, C(s) {Coerce, Not coerce}, what action a A (s) R she takes if she chooses to coerce, and what message m(s) {0, 1} she sends if she chooses not to coerce. The individual s posterior beliefs conditional on receiving message m are given by Pr I (θ =1 m) andapurestrategyoftheindividualspecifiesan action a I (m) R for each message m. As before, we use the solution concept of PBE, in which the authority maximizes her expected utility for each signal given the individual s strategy; the individual maximizes his expected utility given his beliefs after each message; and the beliefs satisfy Bayes rule whenever possible. D.1.3 Nature of disagreement We analyze this model under two different assumptions on the nature of disagreement between the individual and the advisor: Conflicting preferences. Under conflicting preferences, the individual s preferences are biased relative to those of the advisor, i.e., b(θ) = 0 for at least one value of θ; however, the parties have the same opinions, i.e., π I = π A π. In the main body of the paper we assume that preference biases are independent on the state: b(0) = b(1) = b. 39 Conflicting opinions. Under conflicting opinions, the parties have fully aligned preferences, i.e., b(θ) = 0; however, they have different opinions, i.e., π I = π A. 39 This merely simplifies the exposition and makes the underlying intuition more transparent; as we show in the Appendix, all our results hold in the more general case of conflicting preferences. 57

18 D.2 Analysis: the altruistic advisor In this section, we study communication between the individual and his altruistic advisor. The stages of this game are summarized in Figure 6 below. In particular, we analyze how the possibility to sustain truthful communication is influenced by the advisor s regard for the individual. Because this crucially depends on the nature of disagreement between the individual and his advisor, we consider each case in turn in order to isolate the mechanisms. Figure 6: Stages of the setting with altruistic advisor. D.2.1 Conflicting preferences Because the parties share the same prior beliefs, π I = π A π, they hold the same posterior beliefs p(s) =Pr(θ =1 s) aboutthestateoftheworld, conditional on the signal s, p(1) = p(0) = πγ πγ +(1 π)(1 γ), π(1 γ) π(1 γ)+(1 π)γ. Under perfect information, when the state of the world θ is known, the individual and the advisor prefer different actions because b = 0. This disagreement carries on to the case of imperfect information when the individual and the advisor have the same beliefs about the state of the world. We analyze the game backwards. In stage 3, the individual observes the message m, forms his posterior belief Pr(θ =1 m), and chooses the action that maximizes his expected payoff: a I (m) =argmax a R E(U I(a, θ) m). Because the loss function is quadratic, the individual s maximization problem has a straightforward solution: a I (m) =Pr(θ =1 m)+b; that is, the individual 58

19 optimally chooses the action that matches his expected state of the world plus the bias b. If the individual believes that the advisor reports truthfully at stage 2, i.e., that m(s) =s, then the individual optimally chooses a I (m) =p(m)+ b. In this case, a FRE exists whenever the advisor s incentive compatibility conditions for truthful reporting are satisfied. Now consider stage 2. For the advisor to report a signal s truthfully, inducing the action a I (m = s) mustgenerateagreaterexpectedpayoff than inducing the action a I (m = s). First, suppose that the advisor obtained the signal s =1instage1. Inthiscase,herposteriorbeliefthatθ =1isp(1), and her posterior belief that θ =0is(1 p(1)). The advisor s expected utility from sending m =1andinducingactiona I (1) = p(1) + b is given by E [U A (a I (1), θ) s =1] = p(1) (a I (1) 1) 2 + ϕ(a I (1) 1 b) 2 (1 p(1)) a I (1) 2 + ϕ(a I (1) b) 2. If the advisor instead lies, i.e., sends the message m =0,sheincursthecost c and induces the action a I (0) = p(0) + b<a I (1). In this case, her expected utility is given by E (U A (a I (0), θ) s =1) = p(1) (a I (0) 1) 2 + ϕ(a I (0) 1 b) 2 (1 p(1)) a I (0) 2 + ϕ(a I (0) b) 2 c. Hence, after getting the signal s = 1, the advisor prefers to send a truthful message over lying if and only if the following inequality (denoted (TT1 pr ), where the subscript pr indicates the conflicting preferences setting) is satisfied: ϕ 1+ 2b p(1) p(0) c (p(1) p(0)). 2 (TT1 pr) Second, suppose that the advisor gets the signal s = 0. An analogous analysis leads to the following incentive compatibility condition for truth-telling ϕ 1 2b p(1) p(0) c (p(1) p(0)). 2 (TT0 pr) Combining (TT1 pr )and(tt0 pr ) yields that altruism improves communication when the parties have different underlying preferences. In particular, truthful communication can be sustained when the level of altruism ϕ is sufficiently high. This result is formally stated in the following proposition: 59

20 Proposition 5. For any bias b and lying cost c there exists a threshold ϕ(c, b) s.t. a fully revealing equilibrium exists if and only if ϕ ϕ(c, b). 40 Clearly, if b =0,theindividualandtheadvisorpreferthesameaction, and truthful communication can be sustained. When the parties have different preferences, however, communication may break down. If so, strengthening the level of altruism eventually restores truthful communication, because altruism mitigates the preference conflict. To gain intuition for the result, consider the action preferred by the advisor, conditional on getting the signal s, a A (s): a A (s) = 1 1+ϕ p(s)+ ϕ 1+ϕ (p(s)+b). This is a weighted sum of the preferred action of a non-altruistic advisor, p(s), and the preferred action of the individual, (p(s) +b). For her own sake, the advisor would like the action p(s) to be implemented. Nevertheless, when she cares about the individual, her optimal action also reflects that he is better off with the action (p(s)+ b). The stronger the advisor s care for the individual, the more she internalizes the individual s preferences, and the closer is a A (s) to a I (s). Clearly, for a high enough level of altruism, truthful communication is attainable. To better understand the exact form of the truth-telling conditions, consider the cheap-talk case with a zero cost of lying, c =0. Thesymmetric quadratic loss function yields that the advisor wants to induce an action as close to her own preferred action, a A (s), as possible. If the individual s bias is positive, b>0, then the advisor always reports the signal s =0truthfully, because a A (0) = a I (0) + ϕ b is closer to a 1+ϕ I(0) than to a I (1) >a I (0). If she obtains the signal s =1,theadvisorreportsittruthfullyifandonlyifaction a A (1) is closer to a I (1) than to a I (0); that is, if a I (1) a A (1) a A (1) a I (0). This condition can be rewritten 2b a 1+ϕ I(1) a I (0), which is equivalent to (TT1 pr )forc = 0. Analogously, if the bias is negative, b<0, the advisor always reports s =1truthfully,andrevealss = 0 if and only if the incentive constraint (TT0 pr )issatisfied. Corollary 1. The threshold ϕ(c, b) decreases with c and increases with b. The truth-telling threshold has intuitive properties. First, a higher cost of lying c makes truthful reporting more attractive for the advisor. When a FRE 40 This result holds in the general case of state dependent preferences, i.e., when b(0) = b(1). 60

21 can be sustained for a larger range of altruism levels, the level of altruism necessary to induce truth-telling, ϕ(c, b), decreases. Second, a more severe bias b intensifies the preference conflict, which increases the advisor s relative attractiveness of misreporting her signal. Consequently, the level of altruism required to mitigate the preference divergence, ϕ(c, b), increases. D.2.2 Conflicting opinions When the individual and the advisor have different opinions, π I = π A, they have different posterior beliefs p i (s) =Pr i (θ =1 s) giventhesamesignals: p i (1) = π i γ π i γ +(1 π i )(1 γ),p i(0) = π i (1 γ) π i (1 γ)+(1 π i )γ, i {I,A}. In terms of preferences over outcomes, the individual is fully aligned with the advisor, i.e., b =0,whichimpliesthattheyhavethesamematerialpayoffs. Their resulting utilities are given by: U I (a, θ) = (a θ) 2, U A (a, θ) = (a θ) 2 ci {m=s} ϕ(a θ) 2. Under perfect information about the state of the world, there is no disagreement between the parties; they prefer the same action, a = θ. In contrast, under imperfect information, the individual and the advisor prefer different actions even if they have the same signal s about the state of the world, because they interpret this signal in light of their respective (different) priors. In our framework, information is always imperfect, since the signal s is not fully informative (γ < 1). We analyze the game backwards. In Stage 3, given the individual s belief about the state of the world, Pr I (θ =1 m), he optimally chooses the action that matches this expected state of the world, a I (m) =Pr I (θ =1 m). If the individual believes that the advisor reports truthfully at stage 2, i.e., that m(s) =s, then the individual chooses action a I (m) =p I (m). The necessary and sufficient conditions for a truth-telling equilibrium to exist thus coincide with the advisor s incentive compatibility conditions for truthful reporting. Now consider stage 2. First, suppose that the advisor obtained the signal s =1instage1. Inthiscase,herposteriorbeliefthatθ = 1 is p A (1), and her posterior belief that θ =0is(1 p A (1)). Given the individual s strategy, sending m =1inducesactiona I (1) = p I (1), while sending m =0induces 61

22 action a I (0) = p I (0) <a I (1). m =1isgivenby The advisor s expected payoff from sending E A (U A (a I (1), θ) s =1) = p A (1) (a I (1) 1) 2 + ϕ(a I (1) 1) 2 (1 p A (1)) a I (1) 2 + ϕa I (1) 2, where the subscript A on her expectation reflects that the advisor evaluates the expected utility using her posterior about the state of the world, whereas the individual chooses the action that is optimal given his posterior. The advisor s expected payoff from sending m =0isgivenby E A (U A (a I (0), θ) s =1) = p A (1) (a I (0) 1) 2 + ϕ(a I (0) 1) 2 (1 p A (1)) (a I (0)) 2 + ϕa I (0) 2 c. The advisor s incentive compatibility condition for truthful reporting when s = 1(denoted(TT1 op ), where the subscript op indicates the conflicting opinions setting) is given by the inequality E A (U A (a I (1), θ) s =1) E A (U A (a I (0), θ) s =0), which can be rearranged as a I (1) + a I (0) 2 p A (1) + c τ (1 + ϕ), (TT1 op) where τ 2(a I (1) a I (0)). An analogous calculation yields the truth-telling condition when the advisor gets the signal s =0,andcombiningthesetwo conditions yields that a fully revealing equilibrium can be sustained if and only if p A (0) c τ (1 + ϕ) a I(1) + a I (0) 2 p A (1) + c τ (1 + ϕ). (TT op) As we shall see, this condition yields that greater level of altruism worsens the prospects to achieve truthful communication. In particular, if truthful communication can be sustained, raising the level of altruism may eventually destroy it; and if truthful communication cannot be sustained, raising the level of altruism cannot help. Before stating this result formally, we develop its intuition, starting from the special case when c =0. Then,(TT op )reduces to p A (0) a I(1) + a I (0) p A (1). (TT 2 op) Since a I (1) = p I (1) and a I (0) = p I (0), this condition identifies the set of (π I, π A ) for which a FRE can be sustained, given γ. Figure 7 below displays this region when γ =

23 The top solid line represents the advisor s truth-telling condition when getting the signal s =1: shecommunicatess =1truthfullyforall(π A, π I ) below this line. Similarly, she communicates s =0truthfullyforall(π A, π I ) above the solid lower line. Hence, a FRE can be sustained in their intersection, which we denote by TT. The location of TT illustrates that truthful communication requires the priors of the individual and his advisor to be sufficiently similar. Along the 45 -line, their priors are identical; hence, the action that the individual chooses if she gets signal s, a I (s) =p I (s), coincides with the action that the advisor desires him to take, a A (s) =p A (s). Everywhere else, their priors differ, so a I (s) = a A (s). Nevertheless, the advisor prefers to tell the truth so long as a A (s) isclosertotheactionthatatruthfulreportinducesthanitistothe action induced by a false report. To see this formally, we can re-write the right inequality in (TT op) asa I (1) a A (1) a A (1) a I (0), using the fact that p A (1) a A (1). The advisor reports the signal s =1truthfullysolongas the action that she wants the individual to take, a A (1), is closer to the action that the individual chooses under truthful reporting, a I (1), than to the action that he chooses if the advisor lies, a I (0). This obtains because the advisor s expected loss function is monotonic in the distance between the action that she prefers, a A (1), and the action that the individual takes, a I (m). When the priors are so different that the advisor prefers the action that he induces by lying to the action that he induces by telling the truth in some state of the world, the FRE breaks down. For all (π A, π I )belowthelower solid line, the advisor is considerably more convinced than the individual that the state of the world is θ =1(fromex-anteperspective). Inthiscase,the advisor would communicate truthfully when getting the signal s =1;however, when getting the signal s =0,shewouldprefertolieandsendthemessage m =1. Intuitively,theadvisorpreferslyingovertruth-telling eventhough there is no preference conflict because she believes that the individual is so misinformed about the true prior that the individual will take a worse action if the advisor sends a true message than if she lies. A similar logic applies to the region above the top solid line, where the FRE breaks down because the advisor would report the signal s =0truthfully,butliewhens =1. The shape of TT does not depend on the advisor s care for the individual, ϕ. Intuitively, this is because altruism does not influence whether the advisor prefers the action that is induced by telling the truth, a I (m = s), or the action that is induced by lying, a I (m = s). Rather, the level of altruism influences how much the advisor suffers from the implementation of an action that deviates from her own preferred action, a A (s). Indeed, when the advisor 63

24 Figure 7: Truth-telling incentives for different priors (π A, π I ). cares about the individual, on top ofher own material expected loss the advisor also internalizes a share of the disutility that she expects the individual to suffer from his erroneous choice of action. We now consider the case when lying entails a cost, c>0. In this case, truth-telling can be sustained for a larger set of (π A, π I ). Formally, this is immediate from the truth-telling condition, (TT op ): for all (π A, π I )suchthat the advisor is indifferent between lying and telling the truth when c =0 i.e., along the boundaries of TT she strictly prefers to tell the truth when c>0. Intuitively, when lying entails no cost, the advisor would like to lie whenever she believes that lying induces the individual to take a better action than does telling the truth. However, when the advisor is averse to lying, she weighs the cost of incurring a lie against the cost of inducing the individual to take (what she believes to be) a suboptimal action. Clearly, in the presence of a lying cost, she will be more prone to reveal the true signal. The above discussion yields that there exists a nonempty set of (π A, π I ) such that a FRE can be sustained in the presence of lying cost c, but not when c =0. WerefertothisregionasT (ϕ,c). The dotted lines in Figure 7 plot the region T (ϕ,c)forϕ =0andc =0.2. The shape of T (ϕ,c)illustratesthat a FRE always exists when the individual s prior is close to zero or one. This is because the benefit of lying is increasing in the distance between a I (s = m) 64